The particles doesn't move with emitter
The reason why your sprinkles aren't moving with the icing is that the cache for the Particle Emitter is outdated. The cache essentially stores the data for the particle emitter at a certain frame so the computer doesn't have to constantly recalculate it.

I'm not sure if this is a feature or a bug, but when you're moving the donut (which then moves the icing because of parenting), the sprinkles are not getting updated. However, if you select the icing, press G to move, and press the RMB to cancel, the sprinkles will be moved to the proper location. This may suffice as a workaround in your case.

You can also bake the particles as Nate suggested. Baking means that Blender will precalculate the particles' positions, sizes, etc for all of the frames. You can do so by going to the Particle Properties menu, opening Cache, and selecting Bake, Bake All Dynamics or Calculate to Frame. These will take a bit longer to process since Blender will be recalculating the particles for all the frames in the animation even though you haven't touched that yet. So for now, you may just want to stick with the first workaround.